Network doesn't converge with ReLU or Leaky ReLU, but works well with sigmoid/tanh
I don't think it is dead ReLU units as a main cause, although they may be happening as part of the NN failing.

The NN architecture is too complex for the given task (too deep, too many neurons) and that means that any problems you have with other design choices will tend to get amplified. It could be that your NN is close to diverging on the given data and architecture, and that sigmoid is more resilient to it. 

I'd suggest the following changes:


Dropping the learning rate, try 0.01 or even 0.001
Normalising the two input features. NNs like to work with data that is mean 0, standard deviation 1, although there is some flexibility here, your values ranging from 0 to 6 are probably starting to cause minor problems
Look at standard initialisation routines for weights, available within TensorFlow framework, such as Glorot uniform. Your random -1 to +1 is probably too high a range for the given network, and NNs - especially "deep" NNs with 3+ hidden layers - are very sensitive to how initial weights are set.
Simplify the network architecture a little. Five hidden layers and 200 neurons seems a bit much for your goal of over-fitting to this small data set. Try something like 3 hidden layers and 50 neurons.
Your output layer and loss function are designed for a regression task, but you mention that the goal is to identify classes. You need a softmax layer and multiclass log loss for predicting exclusive classes.